Form Approved
REPORT DOCUMENTATION PAGE
OMB No. 0704-0188
Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden to Washington Headquarters Service, Directorate for Information Operations and Reports. 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302, and to the Office of Management and Budget, Paperwork Reduction Project (0704-0188) Washington, DC 20503.
PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. 2. REPORT DATE
3. DATES COVERED (From - To)
1. REPORT DATE (DD-MM-YYYY)
15-02-2002:
Final Technical Report
1996-1999
5a. CONTRACT NUMBER
4. TITLE AND SUBTITLE
Brittle to Ductile Transition in Cleavage Fracture of Alpha-Iron: Experiments and Modeling of Mechanisms
N00014-96-1-0629 5b. GRANT NUMBER
N00014-96-1-0629 5c. PROGRAM ELEMENT NUMBER
5d. PROJECT NUMBER
6. AUTHOR(S)
Ali S. Argon Professor of Mechanical Engineering
5e. TASK NUMBER
5f. WORK UNIT NUMBER
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)
Massachusetts Institute of Technology 77 Massacusetts Avenue Cambridge, MA 02139
20020322 145 10. SPONSOR/MONITOR'S ACRONYM(S)
9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)
Office of Naval Research, Materials Science Prg ONR Code 331 800 N. Quincy Street, Balston Tower 9 Arlington VA 22217-5000
ONR 11. SPONSORING/MONITORING AGENCY REPORT NUMBER
99R00006-01
12. DISTRIBUTION AVAILABILITY STATEMENT
ONR Program Officer Dr. George Yoder; Administrative Grants Office; Director NRL; DTIC Othrewise Unlimited 13. SUPPLEMENTARY NOTES
14. ABSTRACT
In this research the mechanisms of the brittle to ductile transition of fracture initiated primarily by crack tip processes under both quasi-static as well as dynamic conditions was studied experimentally, and mechanistically modeled. Additional experiments and modeling included the role of grain boundariesin resisting cleavage cracking among grains in Fe-3%Si alloy, also associated with modeling. 15. SUBJECT TERMS
Brittle to Ductile Fracture Transitions; Modeling Crack-tip Plastic Response; Crack Arrest by Grain Boundaries 16. SECURITY CLASSIFICATION OF: b. ABSTRACT c. THIS PAGE a. REPORT
U
U
U
17. LIMITATION OF ABSTRACT
U
18. NUMBER OF PAGES
19a. NAME OF RESPONSIBLE PERSON
Ali S. Argon, Princi
P-
Tnvpsf
19b. TELEPONE NUMBER (Include area code>
(617) 253-2217
Standard Form 298 (Rev. 8-98) Prescribed by ANSI-Std Z39-18
OFFICE OF NAVAL RESEARCH FINAL TECHNICAL REPORT
for
GRANT N00014-96-1-0629 PR Number 99R00006-01
BRITTLE TO DUCTILE TRANSITION IN CLEAVAGE FRACTURE OF ALPHAIRON: EXPERIMENTS AND MODELING OF MECHANISMS
Ali S. Argon Professor of Mechanical Engineering Massachusetts Institute of Technology 77 Massachusetts Avenue, Cambridge, MA 02139
February 15, 2002
Reproduction in whole, or in part, is permitted for any purpose of the United States Government.
This document has been approved for public release and sale, its distribution is unlimited.
BRITTLE TO DUCTILE TRANSITION IN CLEAVAGE FRACTURE OF ALPHA-IRON: EXPERIMENTS AND MODELING OF MECHANISMS I.
Summary of Research Carried out under the Program
The research results reported here in this final technical report are a continuation of an earlier program supported by the ONR Mechanics Division under Contract #N00014-92J-4022 where the emphasis was on an experimental study of the brittle-to-ductile transition of fracture in silicon single crystals by thermal crack arrest and on modeling of dislocation emission from crack tips. Similarly, the research initiated under the ONR Program reported here, supported by the Materials Science Division and studying the role of grain boundaries in the crack growth resistance is now being continued under NSF support. The principal thrust in the completed research program under ONR support was on two experimental studies. One thrust was the completion of earlier research on the study of brittle to ductile fracture transition in silicon single crystals as stated above, while the second thrust was on a study of the role of grain boundaries in the enhancement of cleavage crack-growth-resistance in Fe - 3 % Si alloy. The principal results are summarized in the three sections below: II.
Research Report
1.0 Plastic Zones of Thermal Crack Arrest in Silicon. A limiting form of a Brittle-to-Ductile (BD) transition in intrinsically brittle covalent solids is by crack tip shielding resulting from the development of a plastic zone initiated from a crack tip in an initially dislocation free crystal. Where the BD transition temperature is governed by the kinetics of slip-induced-shielding. To account for this effect in dislocation-free silicon crystals the well known crack tip stress relaxation model of Riedel and Rice (1980) for creep deformation was adopted successfully through the use of fundamental crystal plasticity data on silicon to account for the shift of the BD transition temperature on loading rate in Si and to predict the dislocation content of the crack tip arrest zone. This represents the first realistic modeling of the mechanism of crack arrest in Si, and explains the experimental observations very well. 2.0 Experimental Study of the Crack Arrest Zones in Silicon. The main experimental study was on crack arrest in Si single crystals for cracks propagating up a temperature gradient. These verified the rate dependence of the arrest phenomenon and permitted the determination of the activation energy of crystallographic
slip processes from the crack velocity-dependence of the BD transition temperature. Detailed observations of the types and distributions of the dislocations in the plastic zones by combined etch pitting and Berg-Barrett type x-ray topographic imaging of dislocations verified the criteria for the selection of particular slip systems on the basis of ease of nucleation of dislocation embryos at the crack tip from cleavage ledges. 3.0 Role of Grain Boundaries in Cleavage Cracking-Resistance in Fe-3% Si bi-crystals. In this experimental investigation and associated modeling the role of grain boundaries in impeding cleavage crack propagation was studied in a statistically relevant sampling of the behavior of large bi-crystals of Fe-3% Si harvested from a large ingot donated by the Allegheny Ludlum Steel Co. In the experiments the temperature and latticemisorientation-dependence of AK that is required to propagate a cleavage crack across a substantial set of specific large angle grain boundaries were measured to identify the factors that govern the crack propagation resistance of individual grain boundaries. The measured levels of resistances AK resulted partly from additional cleavage plane area produced in the neighboring grain and partly from the plastic-tearing-work bridging these separated cleavage strips. An associated model demonstrated that these resistances are far more importantly affected by the twist misorientation across the boundary plane rather than the tilt misorientation. These results were used to guide a more global study of cleavage cracking-resistance of a field of randomly misoriented grains in a polycrystal. All models quite successfully related to the experimental measurements. Much of this investigation that had been started under ONR support has been carried out under NSF support. III.
Publications Resulting from the ONR Program 1. 2.
3. 4.
5.
6.
A. S. Argon, "Mechanics and Physics of Brittle to Ductile Transitions in Fracture", J. Engng. Mater. & Technology, 123,1,2001. A. S. Argon and B. J. Gaily, "Selection of Crack-Tip-Slip Systems in the Thermal Arrest of Cleavage Cracks in Dislocation-Free Silicon Single Crystals", Scripta Mater., 45,1287, 2001. A. S. Argon and Y. Qiao, "Cleavage Cracking-Resistance of High Angle Grain Boundaries in Fe-3% Si Alloy", Phil. Mag., in the press. B. Gaily and A. S. Argon, "Brittle-to-Ductile Transition in the Fracture of Silicon Single Crystals by Dynamic Crack Arrest", Phil. Mag., 81, 699, 2001. Y. Qiao and A. S. Argon, "Cleavage Cracking-Resistance of High Angle Grain Boundaries in Fe-3% Si Alloy", Mechanics of Materials, in the press. G. Xu, A. S. Argon and M. Ortiz, "Critical Configurations for Dislocation Nucleation from Crack Tips", Phil. Mag., 75, 341, 1997.
IV.
Personnel Associated with the ONR Program. 1.
2. 3. 4.
A. S. Argon, principal investigator, Quentin Berg Professor of Mechanical Engineering. Massachusetts Institute of Technology, Cambridge, MA B. J. Gaily, Research Assistant, Materials Science and Engineering Department, M.I.T., recipient of Ph.D. Degree in 1999. R. Kappaser, Research Assistant, Mechanical Engineering Department, M.I.T. withdrew after one year in 1997. Y. Qiao, Research Assistant, Mechanical Engineering Department, candidate for Ph.D. degree 2002.
Honors Received 1. ASME Nadai Medal, 1998. 2. ETH-Zürich, Switzerland, Standinger Dürrer Medal, 1999.
V.
Appendices
REPRINTS OF PUBLISHED PAPERS AND PRE-PUBLICATION COPIES OF PAPERS PRESENTLY IN THE PRESS Under Grant N00014-96-1-0629
Mechanics and Physics of Brittle to Ductile Transitions in Fracture1 A. S. Argon Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139
The mechanisms of brittle-to-ductile transition offracture in intrinsically brittle crystalline solids such as structural steel have been of great technological interest for a long time. While much useful phenomenology on this important bifurcation behavior has evolved through material testing and alloy development throughout the period following the large scale fractures in Liberty ships during and after World War II, fundamental mechanistic understanding has been lacking until recent times. Over the past decade or so, a renewed level of interest has resulted in a number offundamental studies of both experimental nature and modeling of crack-tip response which demonstrated a remarkable connection of atomic level processes at tips of cleavage cracks and the macroscopic fracture transitions. These mechanistic connections have not only gone a long way in providing basic rationale for some of the successful empirical practices in alloy design and microstructure control, but clear the way for further advances based on basic atomic level processes governing crystal plasticity. Here we give an overview of some recent developments in this area emanating from our own researches. [DOI: 10.1115/1.1325408] Keywords: Brittle-to-Ductile Fracture Transitions, Cleavage Fracture, Crack Tip Modeling
1
Introduction
Ductile to brittle (D-B) transitions of fracture in steel structures are familiar to most mechanical engineers. Many celebrated cases starting with the famous molasses tank fracture in Boston in the winter of 1919 with its bizzare tales of people and horses drowned in molasses; the rash of welded Liberty ships breaking apart on the high seas in World War II and after; the ever continuing discussions on the fate of the Titanic being perhaps due to the brittleness of its steel; the collapse in earthquakes of elevated highway superstructures that were thought to be ductile, all keep the problem current and scary. Through the surge of research and development after World War II in the U.S. and Britain on the B-D transitions, considerable attention was given to the operational aspects of the problem which resulted in much useful alloy development and material testing procedures. This phenomenological perspective, often inspired by some mechanistic insight, has continued up to the present, introducing, more recently, probabilistic aspects of the D-B transition based on the triggering processes that initiate brittle response in structures undergoing ductile behavior. While these developments have resulted in unquestionable improvements in practical applications, they have shed little light on the fundamental governing processes that have wide ranging applications to all intrinsically brittle solids including e.g., ice. Here we will take a brief excursion into some of these fundamental developments with particular emphasis on our own research. 2
Intrinsically Ductile and Intrinsically Brittle Solids
An important insight into the bifurcation phenomenon of fracture came with Kelly et al. [ 1 ], and more specifically with Rice and Thomson [2] who conceived of a fundamental behavior pattern for a theoretical criterion establishing which materials are intrinsically ductile and not subject to a D-B transition, and those 'The 1998 ASME Nadai Medal Award Lecture. Contributed by the Materials Division for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received by the Materials Division June 19. 2000. Associate Technical Editor: D. L. McDowell.
that are intrinsically brittle that can exhibit brittle behavior at low temperatures and high strain rates. In the Rice and Thomson scenario if an atomically sharp mode-I crack in a solid, containing no other imperfections, can emit dislocations from its tip that can then freely multiply before the crack can propagate by cleavage, the solid is identified to be intrinsically ductile under all conditions. Crystalline materials capable of such response are few and are restricted primarily to the metals Ag, Al, Au, Cu, Ni, Nb, Pb, Pt, and Ta, and to some types of amorphous metals. According to this classification all other metals, semiconductors and compounds, including inorganic glasses and possibly all polymers are intrinsically brittle. The latter possess important energy barriers to dislocation emission from crack tips or to initiation of other forms of plasticity in the case of amorphous solids. Nevertheless, they can exhibit ductile behavior above a certain temperature, for a given rate of loading, where these energy barriers can be overcome. It is the mechanisms of fracture transitions in such intrinsically brittle solids ranging from metals to semiconductors to polymers that we have investigated over the years that will be the subject of this paper. Here we will concentrate only on the processes in crystalline solids that are the best understood, and particularly in iron and silicon which represent two special limiting forms of this type of behavior. 3
Ductile to Brittle or Brittle to Ductile Transitions
While the principal familiarity in engineering practice in structural applications is with the D-B transitions, the fundamental aspects are best pursued by considering the reverse process that permits an intrinsically brittle solid to behave in a ductile manner. In one of the more searching early studies on polycrystalline steel, Hahn et al. [3] noted that in smooth bar experiments with a low carbon steel in the D-B transition range of low temperatures the eventual brittle behavior was ushered in by the appearance of grain sized microcracks. These were produced mostly by intersections of deformation twins inside grains; or by cracked carbides. The steel was noted to continue to be ductile when these microcracks were arrested by grain boundaries and rendered ineffective
Journal of Engineering Materials and Technology Copyright © 2001 by ASME
JANUARY 2001, Vol. 123 / 1
by blunting. A transition to brittle behavior was observed at lower temperatures and higher flow stresses when such cleavage microcracks broke through grain boundaries. This specific role of grain boundaries that has not been studied further in any detail is now under investigation by us [4] in special bi-crystal specimens where cleavage cracks are probing grain boundaries of different inclination separating grains having different angles of twist and tilt relative to each other. These experiments are indicating that while additional work of fracture is required for cleavage cracks to break through grain boundaries in a variety of ways involving different kinematical forms, the principal role of the boundary is to stop the cleavage crack for a certain length of time. This permits thermal processes to initiate plastic behavior at the crack tip that can eventually render it ineffective by shielding or blunting. This indicates that the basic mode of the fracture transition is one from brittle-to-ductile (B-D) behavior that involves a crack-tipinitiated thermally-assisted process of emission of dislocations followed by their rapid multiplication in the overstressed crack tip environment. While it might be expected that plastic response in the background, away from the crack tip, can also produce beneficial shielding of the crack tip, this is usually far less effective for several reasons. First, aging often immobilizes the background dislocations making them difficult to displace. Second,- stresses drop sharply away from the crack making such displacements less likely. Thus, in what follows we concentrate on the B-D transition under two different but complementary histories based on cracktip-initiated plasticity. First, we consider stationary mode-I cleavage cracks undergoing a transition from a brittle cleavage response to a tough one with increasing temperature dependent on loading rate. Second, we consider brittle cleavage cracks propagating up a temperature gradient at different constant velocities until they are arrested at a temperature dependent on crack velocity. In both cases the remarkable fact is that, while the transition is in all aspects a macroscopic phenomenon, the key element that governs it involves atomic level processes at the cores of dislocations that control their mobility in different classes of materials. In all cases the eventual tough response of the material involves massive generation and motion of dislocations resulting in large plastic strains at crack tips. Nevertheless, two quite different rate controlling processes can be identified for two separate classes of intrinsically brittle solids based on their dislocation core properties. In the first category are all the BCC transition metals which in pure form exhibit a strong stress dependence of dislocation velocity [5], which has been related to a single energy barrier to the motion of a dislocation line from one lattice potential valley into the next, requiring the nucleation of only a double kink with no further barrier present to impede the motion of these kinks along the dislocation lines, as depicted in Fig. 1 (a). Metals in this category should include Fe, Cr, Mo, W. In these metals the process that triggers the plastic response is thought to be primarily the formationof embryos of fresh dislocations from the crack tip, which can then freely undergo a profuse form of multiplication in the highly stressed crack tip environment to produce the required densities of dislocation line for the development of large plastic strains without further impediment. In the second category are most of the rest of the intrinsically brittle solids in which the motion of the dislocation line from one lattice potential valley into the next, requires not only the nucleation of a double kink but also the motion of the kinks along the dislocation line which is subject to further substantial secondary energy barriers, leading to very sluggish dislocation motion that is only weakly stress dependent. In these materials, among which Si has been widely studied, the final transition to ductile behavior is controlled by the mobility of the groups of dislocations away from the crack tip as depicted in Fig. 1 (b). Nevertheless, even in these materials, experiments indicate that the dislocations that produce the large plastic strains are those that have the lowest energy barrier to their formation at the crack tip. This rather startling behavior that such a major macroscopic 2 / Vol. 123, JANUARY 2001
inclined slip plane dislocation embryo crack
trains of dislocations moving away from crack tip sluggishly
Fig. 1 Configurations of crack tip processes leading from brittle to ductile behavior: (a) formation of a dislocation embryo which can expand freely to result subsequently in unimpeded intense dislocation multiplication as is approximately the case in BCC transition metals; (b) emission of a train of dislocations, sluggishly moving away from the crack tip in materials with high lattice resistance to kink motion along dislocations such as in Si and compounds
transition from a brittle to ductile form, that is known to be affected by microstructure can, nevertheless, ultimately be controlled by atomic level processes at the cores of dislocations is the major finding of the recent fundamental studies. In what follows we take some brief excursions into these recent developments in modeling and review some key experimental results. 4
Modes of Dislocation Nucleation From Crack Tips
Based on expectations, three basic modes of dislocation emission from tips of cleavage cracks were considered in the modeling studies [6,7]. These are illustrated in Figs. 2(a)-2(c) and consist of nucleation of dislocation embryos on inclined slip planes, going through the crack front, on oblique planes intersecting the crack front, and finally on microscopic cleavage ledges produced frequently by small perturbations of the local mode-I loading axis away from the normal to the cleavage planes. The method of analysis that had the required flexibility to deal with the expected three-dimensional forms of the critical dislocation embryos was the variational boundary integral (VBI) technique developed earlier by Xu and Ortiz [8] for dealing with arbitrary convolutions of crack fronts involved in crack trapping, arising from tough local heterogeneities forcing brittle cracks to overcome them by bowing around them. In the VBI method the approach is based on the use of continuously distributed densities of curved segments of dislocations of infinitesimal strength, to rigorously construct any required shape of crack front. In its present application the VBI method is used to model the emergence at the crack tip of incipient local nonlinear displacements involved in the production of dislocations as the crack tip stresses approach the ideal shear strength of a perfect crystal. The methodology is precise and has been discussed fully in previous publications [6-9]. In the next section we outline briefly the key aspects of the VBI approach that is pertinent to the determination of the special activation configurations related to the modes of dislocation nucleation illustrated in Figs. 2(a)-2(c). We consider a semi-infinite cleavage crack and an inclined slip plane intersecting the crack front, as depicted in Fig. 3, as a generic possibility. The crack/slip plane system is loaded remotely by a afield. The crystallographic slip plane is chosen to be the most advantageous one for slip. As the driving force increases, an embryonic dislocation forms progressively until it reaches an unstable equilibrium configuration. The energy release rate G, corresponding to this unstable configuration is defined as the critical Transactions of the ASME
Table 1
Material properties for a-Fe [7]
r slip system (l/2)[lll](lT0) (1/2)[111](112) where ix c «,3) is the normalized activation energy given in Fig. 5, while c is the speed of sound and 37=0.5 relates to the temperature dependence of the shear modulus. Then, the TBD of Eq. (5) can be broadly evaluated for the three modes of dislocation emission of Figs. 2(a)-2(c) for a given crack velocity (say v = 1 cm/s for purposes of visualization). Figure 6 shows the result for some appropriate ranges of the parameter a for each of these modes of dislocation emission of Figs. 2(a)-2(c) and indicates that of these modes only the one based on emission from cleavage surface ledges result in B-D transition temperatures in a reasonable range of 280 K, or close to room temperature and roughly what the TBD is considered to be for low carbon steels. Considering that similar conditions should hold for other BCC transition metals such as Cr. Mo, and W, Eq. (5) can provide estimates for these as well, as 1.26, 1.95, and 2.51 multiples, respectively, of the TBD of steel, using appropriate elastic constants and physical properties.3 Clearly, an effective transition from brittle cleavage to tough ductile behavior must be accompanied by massive dislocation acThe published literature on the actual levels of the TBD of BCC metals, particularly when different small concentrations of other elements arc present varies widely and depends on lestins procedure. For a documentation of this on steels sec Parker [16].
Transactions of the ASME
■t- -r-rrtroi
I
T
IU
.
T
-
103 TBD(poIy-Fe) o c
102 r
/ /
//
-
o 3
U
.
C
M
a _o o
H
"O u
t
io
; D-3
10*
10"'
10
3
10'
102
3
(l-v^UJub
Fig. 6 Estimates of the B-D transition temperature for alpha iron for the three modes of dislocation emission from the crack tip in a scenario of arrest of a brittle cleavage crack travelling up a temperature gradient (after reference [7] courtesy of Taylor and Francis Ltd.)
tivity at the crack tip to arrest its propagation by cleavage, and that emission of an embryonic dislocation triggering the process must be at best only a necessary condition. Experimental evidence suggests that in intrinsically brittle solids having good dislocation mobility (i.e., negligible energy barriers to kink motion along dislocation lines) such as in the BCC transition metals of Fe, Cr, Mo, and W the conditions come close to this relatively simple "triggering" process. Nevertheless, there is considerable evidence for both Mo [14] and W [15] that the level of the TBD can be markedly influenced by overall mobility of the crack-tip-initiated dislocations away from the crack tip. In this, both interactions of the released dislocations with solute atoms and with existing dislocation networks appear to play important roles. These additional interactions, which have their own energy barriers to the motion of the released dislocations, can affect the actual transition temperatures. In distinction to these cases of the BCC transition metals, most other intrinsically brittle crystalline solids with sluggish dislocation mobility (due to substantial energy barriers to kink motion along dislocation lines) the emission of dislocations from the crack tip is only a necessary condition for a successful transition to ductile behavior where the sufficiency arises from the sluggishness of dislocation multiplication and motion away from the crack tip. A good example for such behavior is the case of Si which we will discuss in the following sections. Even in these cases, however, we will note that the crack arrest process is still unmistakably governed by the character of the initial dislocation emission process. 6 Brittle-to-Ductile Transition in Intrinsically Brittle Solids With Sluggish Dislocation Mobility 6.1 Fracture Toughness Experiments at Different Temperatures and Different Loading Rates. The B-D fracture transition has been studied widely in silicon single crystals because Si is obtainable commercially in single crystalline form at closely controlled purities, free of dislocations and in large sizes at relatively low cost. In addition, specimen preparation procedures have been extensively developed by the device industry. These experiments carried out over the years by St. John [17], Brede and Haasen [18], Hirsch et al. [19], George and Michot [20], and others, have demonstrated quite clearly the nature of the fracture transition process as a competition between cleavage fracture and processes of sluggish plasticity that oppose it by crack tip stress relaxation. In particular, the meticulous and elegant experiJournal of Engineering Materials and Technology
ments of George and Michot [20] have demonstrated that development of crack tip shielding by emitted dislocations leading to an eventual fracture transition is associated with a complex set of dislocations moving away from the crack tip. When the loading rate K in static fracture toughness experiments results in a more rapid rate of crack tip stress increase than the rate of stress relaxation resulting from the sluggish dislocation activity in the crack tip plastic zones brittle behavior will follow. The opposite holds when the crack tip stresses never reach levels required for decohesion in the plane of the crack. The above competition is readily modeled based on fundamental information on the kinetics of crystal plasticity in silicon studied by Alexander and Haasen [21] and the stress relaxation model of Riedel and Rice [22] around mode-I cracks in creeping solids. Alexander and Haasen showed that in Si single crystals under appropriate conditions of tensile loading the tensile (equivalent) strain rate e, at steady-state flow, is a power function of the tensile (equivalent) stress a given by an expression e=a(o7o-„)3 exp| - —
(6)
where a/of,= 1.34X103MPa"3s-1 and {/=2.2-2.4eV/atom is the activation energy for dislocation glide (by the continued production and motion of kinks along the dislocation lines), when at steady state a mobile dislocation density of 4.9X10" m/m3 is established. Using this constitutive law the crack tip stress relaxation can be modeled through the Riedel-Rice solution developed for alloys responding to a power-law steady-state creep constitutive relation. Such a consideration gives the crack tip tensile stress
— _ZA
Grain BoundaryPrimary Cleavage facets Secondary Cleavage (ridges) (b)
ms &.•&•&*'■*■■:
l
/ r'
Grain • Boimdary
-*.
TT(C-MC)
(b)
a .Ideal curve Actual curve
Ä. PHILOSOPHICAL MAGAZINE
A, 2001,
VOL.
81, No. 3, 699-740
Brittle-to-ductile transitions in the fracture of silicon single crystals by dynamic crack arrestf B. J. GALLYJ and A. S. ARGON§ Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA [Received 16 February 2000 and accepted in revised form 8 June 2000] ABSTRACT
Experiments have been conducted on the brittle-to-ductile transition of fracture in silicon single crystals through the arrest of cleavage cracks made to propagate on the (110) cleavage planes up a temperature gradient. An activation energy of 1.82 eV has been determined for the transition process based on the dependence of the TBD on an averaged crack velocity, inferred from a jerky mode of crack advance. The dislocation patterns in the arrest zones have been studied in detail by a combination of etch pitting and Berg-Barrett X-ray topographic imaging after the arrest. These observations indicated that the plasticity of the entire arrest process is accomplished by slip activity on a set of two symmetrically placed vertical slip planes in which only one type of dislocation was involved. These planes do not have the highest resolved shear stresses but have the advantage of a very low energy barrier to the nucleation of dislocations from crack tip cleavage ledges. A close correspondence was noted between the spacing of dislocation sources along the crack tip and the density of cleavage ledges observable by Nomarski interference contrast on the cleavage surface prior to arrest A homogenized model of crack tip plasticity is presented that is based on the Riedel-Rice model of stress relaxation at tips of cracks in creeping solids which serves to characterize well all nonlinear aspects of the arrest process. The results have also been contrasted with the predictions of a bnttle-to-ductile fracture transition model based on defect mediated melting and were found to be uniformly inconsistent with that model. §1. INTRODUCTION The abrupt transition in fracture from an anticipated ductile form to a brittle form remains a key phenomenon of concern in the structural use of steels at low temperatures. The worst-case scenario is considered to be the triggering of a brittle cleavage mode of crack propagation in a structure undergoing local large-strain plastic flow. Nevertheless, the complementary point of view of the arrest of brittle cleavage cracks by reduction of crack tip driving forces or by means of local microstructural toughening has been recognized as being of equal fundamental importance in understanding the processes involved in fracture transitions. The early experimental study of Hahn et al. (1959) on coarse-grained E-Steel demonstrated that in tThis paper is dedicated to the memory of Professor Peter Haasen, the former director of the Institut für Metalphysik at the University of Göttingen, who was a pioneer in the recent studies of the brittle-to-ductile transitions in silicon. \ Present address; Exponent, Natick, Massachusetts, USA. § Author for all correspondence. Email:
[email protected] Philosophical Magazine A ISSN 0141-8610 print/ISSN 1460-6992 online © 2001 Taylor & Francis Ltd http://www.tandf.co.uk/journals
700
B. J. Gaily and A. S. Argon
smooth bar experiments, in the fracture transition range of low temperatures, continued tough behaviour results when deformation-induced grain-sized cleavage microcracks could be arrested at grain boundaries. These and more recent experiments (Qiao and Argon 2001) on Fe-3% Si indicate that the principal role of grain boundaries in fracture transitions appears to be to stop the crack for a certain length of time during which thermal processes can result in local plastic relaxations either to shield or to blunt the crack tip. Thus, purely thermally initiated processes of plasticity at crack tips must be considered as playing a key role in the brittle-toductile transitions of fracture in all cases. In the present paper we report an experimental investigation of the processes governing the arrest of cleavage cracks in single crystals of silicon. All previous fracture transition experiments in silicon (St John 1975, Brede and Haasen 1988 Hirsch, et al. 1989, George and Michot 1993) have investigated the brittle-to-ductile transition on static cracks subjected to different rates K of loading at a series of increasing temperatures where the stationary crack either starts to propagate in an unstable manner at lower temperatures or becomes completely shielded by crack-tipmitiated plasticity at or above a certain critical transition temperature and cannot be propagated under the given rate of loading. These experiments have clarified the kinetics of the crack tip shielding process for stationary cracks and probed the specific crystallographic slip processes involved in the shielding (George and Michot, 1993). However, they have not clarified the bifurcation conditions for a crack tip either to propagate by continued cleavage or to be stopped by a progressive evolution of crack tip shielding. Such competing processes involved in the shielding of cracks were observed in LiF by Gilman et al. (1958) and by Burns and Webb (1970a,b). The latter, in particular, observed the formation of plumes of shielding dislocations emanating from propagating cleavage cracks. Similar experiments involving arrest of propagating cracks in silicon single crystals by Brede et al. (1991) and Hsia and Argon (1994) were inconclusive. § 2. CRYSTAL PLASTICITY AT CRACK TIPS IN SILICON For fundamental experiments exploring the purely thermal processes involved in bnttle-to-ductile transitions, silicon is an ideal intrinsically brittle model material of choice. It can be obtained in large sizes in dislocation-free form with precise levels of doping, because of its widespread use in electronic devices which require highly characterized material. Moreover, its crystal plasticity has been extensively studied (Alexander and Haasen 1968, Alexander 1986). The diamond-cubic crystal structure of silicon has the same slip systems as fee metals but has rather more complex dislocation core structures owing to its strong tetrahedral bonding. The core structures of partial dislocations in silicon and their role in the mobility of extended dislocations have been modelled in detail (Bulatov et al. 1995, Cai et al. 2000) in very good agreement with numerous experimental studies of the kinetics of dislocation mobility. Silicon cleaves on both the {111} and the {110} planes with slightly lower surface energies for the latter planes than the former (A:Ic = 0 89 MPam1 * for {110} and 0.93 MPam'/2 for {111} planes (Michot 1988)). This provides alternative choices for either threefold or twofold symmetry for the possible modes of crystal plasticity at the crack tips. In the experimental studies of St John (1975) and those of Brede and Haasen (1988) the {111} plane was chosen for cleavage with the crack propagating m a (110) direction, resulting.in a complex form of crystal plasticity in the crack tip region lacking symmetry with respect to the cleavage plane and making
Brittle-to-ductile transitions in fracture of silicon
701
[110]
[110]
[001]
[110]
[001]
Figure 1. Two crack tip configurations having (110) as the cleavage plane These represent two of the five configurations introduced by George and Michot (1993) which they referred to as e and y respectively. In the present study, only the Q configuration with the highest degree of symmetry was considered. mechanistic analysis difficult. In comparison, George and Michot (1993) considered and explored five different crystallographic configurations of the crack fronts involving both the {111} and the {110} cleavage planes of which both the P and the Q configurations, reproduced in figure 1, utilize cleavage on the more symmetric {110} planes. Of these the P configuration offers a symmetrical set of inclined planes y and 5, in the Thompson notation, that would appear to be well suited to crack tip blunting and a symmetrical set of vertical planes a and ß, in the Thompson notation, as apparently less well suited alternatives for crack tip blunting. The Q configuration in comparison offers a symmetrical set of vertical planes (a and ß) with a better potential for crack tip blunting than the corresponding set of vertical planes of the P configuration and a second symmetrical set of oblique planes (y and 8) that possess similar possibilities for crack tip shielding by dislocation plumes as observed in LiF by Burns and Webb (1970a). Since the y and 5 planes, in the P configuration, also have a high potential for cleavage, they make this configuration less attractive for a study of crack arrest. Therefore, the Q configuration was chosen as the only configuration for the present crack arrest studies. §3. EXPERIMENTAL CONDITIONS 3.1. The crack arrest experiment 3 1.1. Specimen geometry and characteristics An initial attempt to use specially contoured elongated constant stress intensity samples possessing the geometrical advantage for propagation of cleavage cracks at constant velocity by maintaining constant pin displacement rates (Brede et al. 1991, Hsia and Argon 1994) was abondoned owing to its extreme sensitivity to minor perturbations. Instead, a double cantilever-beam (DCB) geometry was adopted, which is intrinsically stable since the crack tip stress intensity decreases with increasing crack length. . Figure 2 shows the basic test configuration of the DCB specimen subjected to a temperature gradient in the crack growth direction, ranging from Tx at the crack tip, being too low to initiate any slip activity, to T2 representing a potential temperature
702
B. J. Gaily and A. S. Argon
d{P/B)/dSj "
(6) K}
The initial coarse estimates of crack growth increments were finally corrected using actual measurements of the final arrested crack length. In many cases the actual crack length was also measured directly by a method of laser light scattering from the edges of the elongating crack, which was described in detail earlier by Brede et cd. (1991), using either a position sensitive detector to record the current crack length or a high-speed (Kodak Ekta Pro) video camera. In such instances the side from which the crack length was measured was free of a side groove. Figure 6 shows an example of jerky crack extension at room temperature through the chevron region of a specimen where the effective ligament thickness at the fracture plane between side grooves
706
B. J. Gaily and A. S. Argon
- '■ --
~j~-l~$-±-4■ end of chevron
Figure 8. Depiction of a relatively idealized jerky crack advance scenario in which the stepwise advancing crack is systematically put into new environments at progressively increasing temperature until it eventually is arrested.
this response time becomes of the order of the residence time of the crack tip at a certain temperature the arrest processes can develop. Under the less ideal conditions of inadequate excess machine stiffness of the present experiments, the crack reached the final arrest position after only one or at most two jumps, leading to considerable uncertainty for the actual time period during which the full sequence of arrest processes were accomplished. In most cases the specimen was unloaded after a major load drop, that had positioned the crack tip in a region of high temperature where a bona fide arrest was expected. While in most cases a substantially completed arrest phenomenon had indeed occurred, in a few cases no dislocation activity could be detected in the subsequent examination of the crack tip region at room temperature, indicating insufficient elapsed time at a high temperature for the arrest processes to develop. In most cases the samples with the arrested cracks were then reloaded to fracture at
Brittle-to-ductile transitions in fracture of Silicon
709
room temperature to reveal the crack front for detailed examination by methods discussed in the following sections. In a few cases the cracked and unloaded specimens were not reloaded to full fracture at room temperature but were sectioned along the median plane, followed by polishing and etching to outline the plastic arrest zone and its dislocation content for analysis as discussed in §§ 5.3 and 5.4 below. 3 5 Material and sample preparation The DCB samples were cut from a single crystal boule of 6 inch diameter and 8 inch length with a [001] axis orientation, donated by MEMC Electronic Materials Inc The crystal was grown by the Czochralski method and was heavily doped with boron to a resistivity of 0.017ftcm. The oxygen content was 117at.ppm with carbon content below 0.5 at. ppm. The heavy boron doping permitted ready cutting by electrical discharge machining (EDM). The relatively high oxygen content resulted in some expected oxide precipitation in the hot zones of the DCB samples and some associated dislocation loop punching as is explained in § 4.4, with, however, negligible consequences for the arrest experiments. The cutting of the specimens to required orientations (always within ±2 ), their polishing, the introduction of side grooves with a 14° included angle chevron at the crack tip, and the drilling of the pin holes have been discussed in detail elsewhere (Gallv 1999) Of these procedures, two proved to be of greater importance than others After being made somewhat oversize by EDM drilling, the pinholes were filled with cylindrical lava-stone inserts, were glued in place with ceramic adhesive and were subsequently drilled and reamed out by conventional machining to give accurately positioned holes for seating the ground tungsten loading pins. After premature dislocation emission from the inadequately finished surfaces of the side grooves in early experiments had resulted in considerable scatter, a detailed and meticulous process of polishing the surfaces of these side grooves was instituted. The finishing steps in this preparation involved chemical polishing with a 22/o KOH bath which exposed selectively the {111} surfaces of the side grooves and sharpened them to a «V shape. This was followed by further exposure to HF, HNO, and CH3COOH, eliminating entirely any spurious dislocation emission from the side grooves by eliminating all EDM-damaged material. Since the polishing of the side groove surfaces rounded the tip of the chevron, a cleavage crack was initiated at the tip by a sharp razor blade. , , . -martte t The properly prepared samples used in the successful crack arrest experiments were essentially dislocation free. The main crack, sharpened by the previously described pre-loading scheme, was the only potential source for dislocations in the crack arrest process. Nevertheless, while the side groove surfaces were smooth and free of residual EDM damage, small gentle variations remained m the ligament oi thickness Bn with a certain density of etching facets resulting from the KOH pohshins step These still contributed to the ubiquitous perturbations, producing jerky crack extension, together with the unavoidable residual 'noise' in the motion of the hydraulic actuator producing the pin displacements. 3 6 Means of examination offracture surfaces and crack tip crystal plasticity Cleavage fracture surfaces were viewed routinely with Nomarski interference contrast microscopy and to a lesser extent examined by atomic force microscopy (AFM) These examinations showed a high density of quite straight but shallow
710
B. J. Gaily and A. S. Argon
surface steps (striations) parallel to the average crack front direction clearly visible in figure 9. AFM showed these to be arranged as a staircase in the same direction. While their origin was not satisfactorily explainable, they were attributed to a possible slight systematic misalignment of the actual {110} cleavage plane with the geometrical median plane of the DCB specimen best suited to crack propagation. In addition to these striations a series of less straight steps of apparently random polarity and nearly perpendicular to the crack front were also observed. These were identified to be cleavage ledges. As is clear from figure 9, they were related in their formation to the striations parallel to the crack front and had a high incidence of having been generated at the time when the striations were formed. The spacing of these cleavage ledges along the striations appeared to be a characteristic of these experiments, with their average spacing remaining close to 5 um. They persisted only for short distances and were smoothed out, only for new cleavage ledges to appear and take their place. From the resolution limits of Nomarski contrast microscopy and AFM measurements the heights of either the striations or the cleavage ledges were estimated to be in the range of 10 nm. While striations parallel to the crack front were considered to be of little consequence, the cleavage ledges perpendicular to them were of considerable consequence in efficient emission of dislocations at crack tips as discussed by Zhou and Thomson (1991) and more specifically by Xu et al. (1997). Those cleavage ledges were attributed to small local variations between the normal to the cleavage crack plane and the mode I axis.
Figure 9 Nomarski interference contrast micrograph of the {110} cleavage fracture surface of a typical specimen showing straight striations parallel to the crack front and cleavage ledges parallel to the crack growth direction which are spawned from the striations and eventually smooth out but are continually replaced with new cleavage ledges.
Brittle- to-ductile transitions in fracture of silicon
711
The nature of the crack tip crystal plasticity at the arrest zone was investigated by two methods that have the overwhelming advantage of both surveying large areas or volumes and providing a considerable level of resolution. They consisted of etch pitting the (110) cleavage fracture surface and the (001) median plane parallel to the external specimen surfaces, which served to map out the nature, extent and dislocation content of the crack tip plastic zones. For these studies the etch employed by Secco d'Aragona (1972) was found to be most reliable to reveal dislocations threading through these two crystallographic planes. While the etch results in a uniform background roughness on a fine scale over the entire surface, the dislocation pits always stand out well and also exhibit a certain conical shape with often eccentrically placed tips suggesting a non-normal emergence angle of the dislocation line relative to the surface. The description of such rich detail which is not of primary interest here can again be found elsewhere (Gaily 1999). The in-depth distribution of the crack tip arrest dislocations was visualized by Berg-Barrett (BB) X-ray topographic imaging of the fracture surface after roomtemperature separation. Topographs of fracture surfaces were taken from several suitable diffracting planes which included symmetric (220) and asymmetric (333), (331) and (620) reflections with information gathered down to a penetration depth limit of nearly 1.0 um. Of these the (333) reflections were of the highest quality and were the most informative. Images were recorded on Ilford L-4 plates with emulsions 10 um thick and silver halide grain size of 0.25um. On the original plates, referred to as negatives, dislocation lines appear as dark lines. All original plates were photographically enlarged by about 20 x and were printed as negatives, preserving the original contrast form. The dislocation line widths varied from 3.6 um for mixed dislocations to roughly 6.7 um for screw dislocations with Burgers vectors parallel to the [110] direction. The BB X-ray topographic imaging method is well developed and has been discussed in great detail in the literature. More detail on its specific use in the present experiments can again be found elsewhere (Gaily 1999). §4. EXPERIMENTAL RESULTS 4.1. Kinetics of crack arrest The details of all the experiments, considered as successful in crack arrest, are listed in table 1 as A. In these, cracks started from the cold end of the specimens and were finally arrested, as expected, in the hot zone near the heating chamber. In a second set of three experiments identified as B the temperature at the hot end of the specimens was not sufficiently high to arrest the propagating cracks, and complete brittle fracture resulted. In two experiments identified as C, cracks jumped into the hot zone where bona fide arrests were expected. However, upon unloading and fracturing the specimens at room temperature, examination of the arrest zone showed no trace of plasticity, indicating that the residence time of the crack in the hot zone was insufficiently long or the temperature in the crack tip region was not high enough for the given pin displacement rate to initiate and develop a shielding process. Finally, in some of the earlier experiments where the anticipated arrest temperatures for higher pin displacement rates required higher temperatures at the hot end, the temperatures at the low end at the initial crack length became too high. This prevented initiation of brittle crack advance. Subsequent examination of the
712
B. J. Gaily and A. S. Argon • .3
CM
s SI
£
u
ca
u ca
ca
j-.
U
s °
CO
CO *-*
ca K
H
{TTN.
Nt m^-« M rn * m O m ON TC '^■ O o ON 00 00 u-i
o m
U ca g p
fli -^
°
0*3.0
cafc ca? cafc ca§ caS
ca
3 O o .3
ca ^
■«•IT-ONO
NO NO >/-> NO IO U~i
E
•oo"
T) 3
■"SB O •— a>
sp 1
s 0 3 P- ca .3 ft CJ
qviN-i-o —< Ö Ö O Ö Ö
© ©
00 ©
CN ©
OO © ©
T3 > p C3
Is .
«u 2 » B .
x> S CO i
(JO
3 ca
o £ ■3 S o
S 8
S «s g u
O
ca
ju
ca
CNO
c oa -^i^ s» c
s gj
ft W
s u
ft
CO
T1 i)
H_
O 'S ft^> 3 O ca o ■ ca ^_ > a ■g-o
tx P
©
ON
621 «
00 ~x NO m
fa' SJSJSt 3 3
U
oa
03
09
co ca "^ m (D
C^H
CA
P
■—< cN m ^a- in
NO
^s 0)
X) ca
" II «j u — r* ^ are in the slip planes and are analogues to r and 6 and, finally, V is the angle between the slip plane and the plane normal to the crack front as seen in figures 3 (a) and (b). The factors Re (gnm) for the resolved shear stresses a„„, were computed with the elastic compliances appropriate for
Brittle-to-ductile transitions in fracture of silicon
723
silicon, that is, su = 0.7685 x KT11 Pa-1, sl2 = -0.2139 x 1(TU Pa-1 and J44 = 1.2563 x 10~nPa_1 (Simons and Wang 1971). The results for the vertical a and ß slip planes are plotted in figures 17(a) and (b), and those on the oblique S and Y planes in figures 18(a) and (b). While the nucleation of CD-type screw dislocations with Burgers vectors parallel to [110] should have been preferred, the BB X-ray topography results were unable to differentiate between these screw dislocations and the 60° mixed dislocations of AC or AD type through a standard extinction contrast analysis. Both families of dislocations exhibited extinction contrast on all the surface diffractions taken. Inspite of the favourable conditions for the nucleation of CD-type screw dislocations, George and Michot (1993) have reported observing only the 60° mixed dislocations on the vertical a and ß slip planes in the plastic zones of cracks in the Q crack tip configuration. Their experiments were performed, however, on samples of only 0.6 mm thickness as opposed to the 3 mm samples used here. When their Burgers vector analysis could be performed, sufficient plasticity must have developed for the crack tip plane strain stress to relax substantially. This is expected to change significantly the resolved shear stresses activating slip on the various slip systems and may be the explanation for their observations. 5.2. Shapes of emitted dislocation loops The resolved shear stresses calculated in § 5.1 can also yield information on the initial shapes of expanding dislocation loops which are of particular interest for the CD screw dislocations on the a and ß planes. To a first approximation, dislocations should expand, at least initially, on their slip planes to a shape along which the force acting to expand them is constant. This shape can be obtained by contours of resolved shear stress acting on the slip plane, parallel to the Burgers vector of the dislocation. Such contours can be determined by solving equation (9) for p( C . Sin(pCOS(p" >+B (10) cos -(cosy) ' " ^
/AXcr
l
;(sin(p 2
where the brackets across the first and second terms on the RHS of Eqn.(10) indicate an orientation average of all possible cases of sampling of grains of random orientation. Rather than develop this expression as given, we consider a model of the above ratio for each grain in the specific case of the polycrystal field considered in Fig. 15 and obtain the cumulative work of fracture for the quasi-statically advancing cleavage front according to the modeled percolation map of Fig. 18. The result of this model, following along the same set of steps developed in Section 4.3 for the determination of the percolation arrow map of Fig. 18, is shown in Fig.22. In the model, the critical penetration depth Axc for break-through of a grain boundary was taken as 10 urn for the very coarse-grained material modeled here. Since in the present case the average grain sizes were in the range of 4mm the ratio Axc/D becomes roughly 0.0025, making the contribution of the second term in Eqn.(10) negligible for such large grain size material. As stated above, for very
-37-
fine grain material, we assume that Axc should become proportional to the grain size, where the second term in the RHS of Eqn.(lO) will make a non-negligible contribution. It was found that if B is in the range from 2.0 to 3.0, the model results produce a very good fit to the experimentally observed fracture toughness.
Parenthetically, we note that the choice of the magnitude of the constant B as about 2.5, and taking 8c = Axc = lOfxm, Ah/D = 0.5 and GiC = 230J/m2, derived from the
average fracture toughness of 7.3MPaVm, measured in a thermal crack-arrest experiment in single crystals of Fe-3%Si alloy (Qiao and Argon, 2002b), would give for the "cleavage-like" shear resistance k a value of about 115MPa. This value of k is in the range of the plastic shear resistance of the alloy under consideration and indicates that the shear fracture process along the bridging grain boundaries could have a substantial plastic shear character, as shown in Fig.21 and very similar to what was considered by McClintock and Clerico (1980).
5. DISCUSSION
38-
5.1 Contribution of grain boundaries to the cleavage resistance
Based on the detailed considerations of the percolation of cleavage fracture across a field of grains in the coarse-grained Fe-2%Si alloy discussed in Section 4.2 and the models of the percolation map and work of fracture developed in Sections 4.3 and 4.4, it becomes possible to state a general expression for the contribution of grain boundaries to the overall work of fracture of a polycrystal in the extreme lower shelf region of fracture, labelled by us as pure cleavage (PC) fracture. The expression we propose is a generalization of Eqn.(lO), benefiting from the numerical model of Section 4.4. For this:, purpose we re-state Eqn.( 10) in its final form below as:
2*29 G
ic
= +oPr^>+B(ii)
cos\]/0cos(pJ0
^VJ (cosy) .
cosV
In Eqn.(l 1) in the first term on the RHS Cs = 2/Jl is a factor correcting for hexagonalshaped grain intersections over the square-shaped grain facets considered in the preliminary rough model; the orientation average of the projection product in brackets is readily evaluated as 1.26. In the second term a = Ji is the ratio of the average grain diameter to the grain edge length of a hexagonal grain and the factor ß = 1.14 accounts
-39-
for the fact that in 14% of cases in the simulation a cleavage crack enters a grain through two faces nearly simultaneously; the orientation average of the term in brackets can be taken as 1.61, as the average value of this factor determined in a previous study of cleavage cracking resistance of bicrystal boundaries (Qiao and Argon, 2002a). The ratio Axc/D in this term remains somewhat elusive since the effective penetration distance Axc across a boundary where the peak resistance is encountered can only be conjectured. We have arbitrarily considered this dimension to be around lOpim for very large grains with £>»Axc. As stated above, however, we consider the ratio Axc/D to remain a constant for D < 10 |im. In any event we note that the second term is the only one with a grain size dependence, which becomes negligible for very large grain sizes and becomes grain size independent for medium to small grain size material. Finally, as discussed in Section 4.4 above, we have chosen the magnitude of B as 3.0 based on the model of the rate of increase of cleavage fracture work across a field of coarse grains as given in Fig.21. This gives the final form of the cleavage resistance of the polycrystals as:
-^ = 1.45 + 3.03(Axr/D) + 3.0 G
(12a)
ic
= 4.45+3.03 (Axc/D)
(12b)
We have written Eqn.(12a) deliberately in an expanded form to indicate that the contribution to the overall cleavage work of the cleavage of individual grain interiors is -40-
substantial and represents roughly one third of the total work of fracture, as should be clear from Fig.21. The second term on the RHS of Eqns.(ll) and (12a) makes a negligible contribution to the total work of fracture for the large grain material that we have considered specifically even though it is this term that governs the selection of the sites for cleavage break-through across grain boundaries. However, because of the tiered nature of the cleavage break-through across grain boundaries described in detail in the earlier bicrystal study (Qiao and Argon, 2002a) where the majority of break-throughs had been noted to follow a regular mode of penetration, only a relatively small overall change of fracture area results as the cleavage front crosses a typical grain boundary. The somewhat more substantial contribution (roughly two thirds) to the overall work of fracture results from the process of cleavage-like shear separation along grain boundaries (Fig.21) between primary cleavage cracking facets in adjoining grains, as had already been concluded by McClintock (1997).
Finally, we return to the exploration of a grain size dependence of the cleavage fracture work. While no important grain size dependence is evident in our model beyond the weak and unimportant grain size dependence represented in the second term of the RHS of Eqns.(ll) and (12a, b), we have examined the specific effect experimentally. This dependence was probed through fracture toughness determinations at -20C in the pure -41-
cleavage range in 8 coarse-grained samples, in which the average grain size varied between 1.5mm and 6.0mm. The result is given in Fig.23. It shows very substantial scatter among the measurements, with no discernable grain size dependence which is consistent with our model in which the only grain size dependent contribution might conceivably arise for grain sizes in the range of 50-100 microns but should be negligible in the coarse grain size range of 1-10mm, and disappear for very small grain size material if the ratio Axc/D reaches a constant value asymptotically, as was assumed might be the case.
5.2 General Observations
The role of grain boundaries and grain size effects in the fracture resistance of polycrystalline metals, particularly steel, has been well appreciated for many decades and has been modeled by Hall (1951) and Petch (1953) among others. Similarly, the effect of grain size on fracture and the beneficial consequence of grain refinement in increasing the brittle strength of steels is also well known (Petch, 1954). Furthermore, the triggering effects of fracturing grain boundary carbides in initiating brittle behavior in fracture transitions have also been studied to some extent (McMahon and Cohen, 1965; Ritchie, et al., 1973; Lin et al., 1986; Petch, 1986), as we had already noted in Section 1.0. Outside -42-
of these general considerations of grain size effects and phenomena related to triggering of brittle behavior in fracture transitions, specific considerations of the influence of grain boundaries on the cleavage cracking resistance have been few. Of these studies the most noteworthy have been those of Gell and Smith (1966), Anderson et al. (1994) and McClintock (1997). In the work of Anderson et al. (1994), the mode of cleavage fracture across hexagonal grains was considered in some detail to develop realistic cumulative probability distributions of fracture resistance in the context of "weakest link" models of such behavior. Gell and Smith (1966) have made one of the earliest observations on the cracking resistance of individual grain boundaries with known tilt and twist components, in hydrogen charged polycrystalline Fe-3%Si alloy and demonstrated that the major contribution to the cracking resistance of a grain boundary comes from its twist misorientation across the boundary rather than the tilt misorientation. This basic observation, also conjectured by McClintock (1997), was demonstrated directly by us in our earlier bicrystal study (Qiao and Argon, 2002a). The most detailed study of the cleavage cracking resistance of polycrystals, with which we will compare our findings is that of McClintock (1997), who has modeled the form of percolation of a cleavage cracking front across an array of randomly misoriented cubic grains. In his study McClintock makes a series of assumptions that have actually been observed, in part, in the present experimental study. These included the following: that a) grains crack by a -43-
cleavage crack entering them from a neighboring grain rather than by the applied stress on the grain, b) a tilt misorientation across a boundary is far less of an obstacle for penetration across a boundary than a twist misorientation; c) normal expectations apply that cracking in re-entrant channels formed by cleaved adjacent grains do not readily extend while exposed corners on crack fronts are favored for cleavage propagation, and d) certain difficult-to-shear boundaries connecting facets of already cleaved grains (recalcitrant grain boundaries) effectively hold back the propagating cleavage front. While some conclusions of the McClintock model such as an excessive tendency of cracking fronts to advance parallel to the front by laterally propagating cleavage "kinks": along the front, was not found in our observations, many other features were quite similar. The single major departure, however, between the predictions of McClintock and our model has been that the fracture toughness in his model exhibits a square-root type of grain size dependence (linear grain size dependence of work of fracture) while no such grain size dependence resulted from our model. This difference results primarily from considering that the cleavage work of fracture of grain interiors is negligible in comparison with the plastic bridging shear work along boundaries connecting primary cleavage facets, and that the latter process is indeed a plastic shearing-off process. In our model the cleavage work of grain interiors makes up fully one third of the overall fracture work and the bridging shear processes are viewed more as a cleavage-like shear fracture -44-
in which the traction drops abruptly to zero after a preparatory constant shear displacement emanating from the cleavage facets as depicted partly in Fig.21 rather than a continuous linear drop characteristic of a pure plastic shearing-off process. Nevertheless, since there are significant uncertainties in both our model and that of McClintock the exact nature of grain size dependence of the cleavage resistance remains unanswered. In any event it is clear that in the very large grain size limit there is no observable dependence which is in support of our model.
5.3 The decarburized 1010 steel
The experimental excursion into the behavior of the 1010 steel was made for the purpose of demonstrating that the findings on the Fe-2%Si alloy could be used in applications to low carbon steels. We note that our findings on the role of grain boundaries on the cleavage fracture resistance of the Fe-2%Si alloy are primarily of a geometrical nature exhibited by the strongly solid solution strengthened single phase bcc material. Thus, the potential applicability of these findings to low carbon steel is appealing. In our experiments with the decarburized 1010 steel where a potentially complicating pearlite component was removed a full demonstration of a parallel behavior to the Fe-2%Si alloy was not quite possible because of the very much reduced level of plastic resistance of the -45-
former shown in Fig.2 that resulted in all fracture measurements on the 1010 steel to be effectively on the upper shelf, even at the lowest test temperature of -150C. This was evident from the large distortions of the DECP specimens at fracture and relegates the fracture work measurements of JjC to being of plane stress type. Nevertheless, much can be concluded from the cleavage fracture surface features shown in Figs.l la-1 lc, albeit all being in the nature of terminal cleavage occuring after very substantial amounts of expanded plastic work in the ligament between the two edge cracks. The fracture surface of Fig.l la at -125C shows a field of cleaved grain facets of a distinctly brittle appearance with only weakly delineated river markings which we would have expected to be quite similar to those observable in the pure cleavage realm at the lower shelf below the ductile-to-brittle (DB) transition temperature, which we expect now to be well in the cryogenic temperature range near liquid nitrogen temperature on the basis of the shift of the plastic resistances and the estimate of the brittle strength of the 1010 steel. The fracture appearance in Fig.lib at -70C, now well above the DB transition, shows still a field of cleaved grains, but with well delineated cleavage river markings giving clear evidence of accompanying plastic distortions resulting from the bridging deformations between grains.
46
The most profound observation derivable from the comparison of the behavior of the Fe2%Si alloy and the decarburized 1010 steel is the dramatic reduction in the conventional ductile to brittle transition temperature by about 400C, from 250C (for the Fe-2%Si) to -150C (for the decarburized 1010 steel) when all hardening agents are removed from the i
:u \
.
'
•;.
■.«.:*
■-.;
•
r
*
.
Fe. We note that this effect if far larger than what might be expected from inspection of Fig.2 and the shift of the temperature dependent plastic resistance curves. This further accentuation of the difference between the DB transitions results from the fact that the J1C measurements for the 1010 steel are of a plane stress type in a DECP specimen of quite inadequate size and shape while the results for the Fe-2%Si alloy were derived from bona-fide plane strain test configurations.
6. CONCLUSIONS
In the lower shelf in Fe-2%Si and in low carbon steels cleavage cracks percolate through grains by selectively going through the most exposed grain boundaries with the lowest penetration resistance and subjected to the largest local crack driving forces. While these grain boundaries govern the overall mode of cleavage crack percolation they contribute little to the overall fracture resistance.
-47-
The overall fracture resistance is derived roughly in one part by the cleavage resistance of individual misoriented cleavage facets in grains and in two parts by the work of fracture along nearly vertical boundaries bridging the primary cleavage facets of adjoining grains by a combination of plastic shear and a "cleavage-like" shear separation.
Both experiments and modeling of crack percolation mechanisms suggest that the overall cleavage fracture resistance due to the tesselation of the cleavage fracture process by the grain boundaries is largely grain size independent, but increases the fracture resistance in comparison with a flat untesselated cleavage process by a factor of close to 3.0.
Experiments carried out on decarburized 1010 steel indicate that, when present, the percolation of cleavage across grains in the upper shelf region is very similar to that in the lower shelf, which has important implications in the triggering of brittle behavior in structures undergoing initial ductile forms of fracture.
48
ACKNOWLEDGEMENT
This research has been supported by the National Science Foundation under grant DMR9906613. We are also grateful to Professor L. Anand for providing the coarse grained Fe2%Si alloy plates and to Professor F. A. McClintock for useful discussions.
49
REFERENCES
Anderson, T. L., Stienstra, D., and Dodds, R. H., (1994), A thermodynamical framework for addressing fracture in the ductile-brittle transition region, in Fracture Mechanics: Twenty-fourth Volume, ASTM-STP 1207, edited by Landes, J. D., et al, ASTM: Philadelphia, p. 186.
Argon, A. S. and Qiao, Y., (2002), Cleavage cracking resistance of high angle grain boundaries in Fe-3%Si alloy, Phil. Mag., in the press.
Birks, N., (1969), Decarburization, the Iron and Steel Inst, London.
Brown, S. B., Kim, K. H., and Anand, K., (1989), An internal variable constitutive model for hot working of metals, Internat. J. Plasticity, 5, 95.
■•>
Gell, M. and Smith, E., (1967), The propagation of cracks through grain boundaries in polycrystalline 3%Silicon-iron, Ada Metall., 15, 253.
Hall, E. O., (1951), The deformation and ageing of mild steel: III - Discussion of results, Proc. Phys. Soc, B64, 747.
Kumar, V., German, M. D. and Shih, C. F., (1981), An engineering approach for elasticplastic fracture analysis, EPRI Report NP-1931, Palo Alto, CA: Electric Power Research Institute.
Lin, T., Evans, A. G., and Ritchie, R. O., (1986), A statistical model of brittle fracture by transgranular cleavage, J. Mech. Phys. Solids, 34, 477.
-50
McMahon, C. J. and Cohen, M., (1965), Initiation of cleavage in polycrystalline iron, Acta Metall., 13, 591. McClintock, F. A., (1997), A three-dimentional model for polycrystalline cleavage and problems in cleavage after extended plastic flow or cracking, in George Irwin Symposium on Cleavage Fracture, edited by Chan, K. S., TMS, Warrendale, PA, p.81.
McClintock, F. A. and Clerico, M., (1980), The transverse shearing of singly-grooved specimens, J. Mech. Phys. Solids, 28, 1.
Petch, N. J., (1953), The cleavage strength of polycrystals, J. Iron Steel Inst, 174, 25.
Petch, N. J., (1954), The fracture of metals, in Progress in Metal Physics, edited by Chalmers, B. and King, R., Pergamon Press: London, vol.5, p.l.
Petch, N. J., (1986), The influence of grain boundary carbide and grain size on the cleavage strength and impact transition temperature of steel, Acta Metall, 34,1387.
Qiao, Y. and Argon, A. S., (2002a), Cleavage cracking resistance of high angle grain boundaries in Fe-3%Si alloy, Mech. Mater., in the press. Qiao, Y. and Argon, A. S., (2002b), Brittle to ductile transition of cleavage fracture by thermal crack arrest in Fe-3%Si single crystals, to be submitted for publication.
Rice, J. R., (1985), First-order variation in elastic fields due to variation in location of a planar crack front, J. Appl. Mechanics, 52, 571.
Ritchie, R. O., Knott, J. F. and Rice, J. R., (1973), On the relationship between critical tensile stress and particle toughness in mild steel, J. Mech. Phys. Solids, 21, 395.
Thelning, K.-E., (1984), Steel and its Heat Treatment, Butterworth: London. -51-
FIGURE CAPTIONS
Fig.l Sketch of the modified stereo microscope for determination of inclination of cleavage planes in individual grains relative to sample reference surfaces.
Fig.2 Temperature dependence of tensile yield stresses in Fe-2%Si alloy (as scaled from data on Fe-3%Si single crystals) and decarburized 1010 steel.
Fig.3 Temperature dependence of the fracture toughness, £"IC, and fracture appearance in coarse grained Fe-2%Si alloy.
Fig.4 SEM micrograph of fracture surface of Fe-2%Si below the PC/MC transition at -20C.
Fig.5 SEM micrograph of fracture surface of Fe-2%Si above the PC/MC transition at 20C.
Fig.6 SEM micrograph of a complex bridging fracture between two parallel cleavage facets inside a grain in Fe-2%Si alloy at -20C.
Fig.7 SEM micrograph of a complex fracture along a recalcitrant grain boundary in Fe2%Si alloy at-20C.
Fig. 8 SEM micrograph of fracture surface immediately above the conventional brittle-toductile transition at around 250C in Fe-2%Si alloy, showing areas of clear ductile dimple fracture and cleavage fracture.
Fig.9 Temperature dependence of J\C plane stress fracture toughness in DECP specimens of decarburized 1010 steel showing the persistence of terminal cleavage fracture above the brittle-to-ductile transition in the upper shelf region. -52-
Fig. 10 Fracture surface of decarburized 1010 steel at -100C, well above its brittle-toductile transition temperature, showing a mixture of cleavage and ductile dimple fracture.
Fig.ll A sequence of SEM micrographs of fracture surfaces in decarburized 1010 steel: a) at -125C; b) at -70C; and c) at -40C. Fig. 12 Various modes of entry of a cleavage front into individual grains: a) penetration into an exposed grain; b) penetration into a grain through a particularly weak boundary; c) and d) cracks entering into grains with an exposed corner.
Fig. 13 Macrograph of a fracture surface of a compact fracture specimen of very coarse grained Fe-2%Si alloy. The vertical thickness dimension is 26mm.
Fig. 14 Orientation relationship of cleavage facets in individual grains, relative to external specimen axes. Fig. 15 The cleavage fracture percolation map through the grains of the sample shown in Fig. 13, showing also two crack arrest fronts A-A and B-B.
Fig. 16 The calculated distribution of local stress intensity factors K\ along the fracture fronts A-A and B-B. Fig. 17 Histogram showing the distribution of numbers of grain edges in the fracture field of Fig. 15, not counting grain edges bounded by free surfaces.
Fig. 18 Computed cleavage fracture percolation map using information on grain boundary resistance determined from Qiao and Argon (2002a).
Fig. 19 a) A schematic view of the types of boundaries overcome by the cleavage front. Boundaries with emanating arrows are those that govern the form of penetration of the -53-
fracture front from grain to grain. Boundaries between 1-2, 5-6, and 2-4, in various measure, require extensive fracture work to bridge primary cleavage facets; b) simplified field in a generic square grain: (1) cleavage work inside grain; (2) work of fracture for going through the principal boundary controlling the percolation process; (3) boundaries requiring large amount of bridging fracture work by plastic shear and cleavage-like separation.
Fig.20 Stages of bridging fracture work: a) initial outline of grain boundary bridging primary cleavage facets; b) preparatory plastic crack tip displacement triggering subsequent "cleavage-like" separation.
Fig.21 SEM micrograph of the surface of separation by shear in a typical boundary connecting two cleaved facets in adjacent grains. Most of the surface shows signs of plastic shearing. Portion identified by an arrow shows a fracture type separation.
Fig.22 Result of ccomputer simulation of overall fracture work for an advancing crack front through the family of grains shown in Fig. 15. Stars indicate accumulated fracture work due to cleavage of grain interiors, open circles indicate cumulative fracture work contributed by the severence of the bridging grain boundaries.
Fig.23 Grain size dependence of critical stress intensity of fracture at -20C in Fe-2%Si alloy.
-54
TABLES
1. Chemical composition of the Fe-2%Si alloy.
2. The tilt and twist misorientations of principal cleavage facets of 78 individual grains in the compact fracture sample of Fig. 15 referred to external axes.
55-
9-
Fixed colimated light source
Stereo microscope
Specimen
Double-angle milling vise
700
°
600
cS
/Twining
500 Fe-2%Si alloy as rescaled from Fe-3%Si
ÖD
C
400
X/l
13
300
200
Decarburizei 1010 steel
100 200
-150
-100
-50
0
50
100
150
Temperature (°C)
2.
100 o Critical stress intensity factor £jC * % strips in mixed-cleavage area with evidence of plastic dissipation n % fibrous in the fracture surface 80
4
60
CO
y
40
8.
^
Po
20 o o
40(%)
#
20
100
150
• •• • -a—m a' -B200 250
0 300
350
Temperature (°C)
3
£
(o
7
sr
80
o
ic
./i
o
o O
O
60 o o
(KJ/m2) 40
S3 o
Fraction of fibrous area in fracture surface
o
V*
100%
* *
20
Fraction of grain boundary with . plastic deformation in MC area 0
-4
-150
-*S-
m-
-100
-50
0 Temperature (°C)
50
100
50% 0% 150
\o
(c)
1/
O
A
D
P
(b)
(a)
A
(c)
D
A
D
(d)
12
iß
O
3 M o
c3 O
8
Di
es o
Crack Propagation Direction
#
rr
_ K(zyK„
a(z)
0.0
100
200
z
ropag£ crack fro t
300
400
(a) Crack front A-A
gating front
400 (b) Crack front B-B
ß
0.25
ö o
•I-H
I 0.15 •!-H
B
•1-H
Ö
o o cd HJ.u;i4i
I! ,'-l?XW nr.7S571
II 4KV7M —\
11.4214?» «71-M
1
n
IM:«.
G/GK-1.86
The saddle-point configuration of a dislocation embryo emitted from the crack tip on an oblique plane in the interior. Fig. 9 6
■
5 -
■
1
'
'
'
'
■
1
\
'
'
'
1
'
'-"
'
1
'
'
-r ■.
|
.
\ oblique plane \ \
I4 X>
D < 3: P
i
\
-
l
\
x
"
i
2 ;
\
1
\ \
...... .xi
0
0.5
\
.......
1.0
1.5
2.0
2.5
G,/GIC Activation energy for dislocation nucleation on an oblique plane in the interior ( and near a free surface ( ).
,
)
tions in our analysis. One-sided configurations, in which dislocation embryos expand primarily on one side of the oblique plane, might conceivably require lower activation energies. However, in view of the results just described, it seems unlikely that this reduction in the activation energy should be sufficient to justify a detailed analysis of one-sided configurations.
354
G. Xu et al.
The above analysis establishes convincingly that nucleation of dislocations on oblique planes in the interior of the cracked solid is most unlikely. A very different conclusion can be reached, however, for this mechanism where the crack front reaches a free surface where no plane strain stress is present, and the resolved shear stresses on the oblique planes become much higher. An estimate of this enhanced nucleation probability on the oblique plane near the surface is easily obtained by rescaling the driving forces in fig. 9, in proportion to the resolved shear stresses on the oblique slip planes near the free surface against those in the interior as (for the geometry of fig. 2 (6))
where GIs and Gjj are respective energy release rates required for initiation of a dislocation near the free surface and in the interior. The result for a-Fe with v = 0-291 is shown as the chain curve in fig. 9, which now suggests almost spontaneous embryo formation near the free surface. This, however, is not true, since in this case at least a partial surface ledge must be produced, which will make the nucleation more difficult, but presumably still much easier than in the interior. An abundance of such nucleation events has been observed by George and Michot (1993).
4.4. Nucleation of dislocations on a cleavage ledge Cleavage surfaces in metallic crystals invariably contain ledges parallel to the direction of crack propagation. These are likely to form when the principal tension driving the crack deviates slightly on a local scale, requiring the crack to make small adjustments along its front. This microroughness of the cleavage surface depends also on the crystallography of the cleavage planes and crack propagation direction as well as on temperature. The height of the observed ledges can range from several atomic spacings to microns. Numerous observations (Chiao and Clarke 1989, Hirsch et al. 1989a, b, George and Michot 1993) have revealed that dislocation nucleation at a crack front is a relatively rare phenomenon associated with crack front heterogeneities. This strongly suggests that ledges are likely sites for heterogeneous nucleation of dislocations (Zhou and Thomson 1991). In what follows we analyse this mechanism as it is likely to operate in a-Fe. Consider a cleavage crack propagating under mode 1 loading. The crack contains ledges of a width of roughly a hundred atomic spacings distributed along its front, as depicted later in fig. 11. The presence of a considerable local mode III stress intensity factor acting on the ledge is expected to promote nucleation. Moreover, the direction of the Burgers vector of the dislocation embryo, which is parallel to the local crack front on the ledge, requires no fresh surface production. In view of the mesh size requirements to resolve adequately the dislocation embryo, a direct simulation of the complete system does not appear possible at present. This difficulty can be sidestepped by the approximate two-scale approach sketched in fig. 10. The distribution of stress intensity factors along the front of the crack, including the ledge, is first calculated by recourse to a linear elastic.analysis. The small stretch of ledge on which the dislocation embryo nucleates is then idealized as a semi-infinite crack subjected to the local stress intensity factors determined in the first analysis. Because of the vastly disparate scales of the ledge and the activation configuration, the results
Dislocation nucleation from crack tips
355
Fig. 10
Dislocation nucleation on a cleavage ledge.
obtained in the manner just outlined should be ostensibly identical with those obtained from a direct simulation. The distribution of stress intensity factors on the crack front can be readily calculated by the boundary element method of Xu and Ortiz (1993). The mesh used in the analysis is shown in fig. 11. As is evident from the figure, two symmetric ledges are included in the mesh. This permits the enforcement of periodic boundary Fig. 11
Cleavage Ledges
Mesh used in the computation of the stress intensity factors along a crack front containing cleavage ledges.
356
G. Xu et cd.
conditions, which greatly facilitates the calculations (Xu and Ortiz 1993). For small width-to-separation ratios, the interaction between the ledges may be expected to be negligible. The calculated stress intensity factors are shown in fig. 12. On the ledge, the dominant stress intensity factors are K,ledge « 0-8IK, and K\\fse « 0-35A",. On the verge of brittle fracture, it therefore follows that K1^ « 0-35A"ic. For mode 111 loading, Rice (1992) has determined the athermal critical condition for nuclcation of a screw dislocation to be G|||cd — 7us — ^~-allied-
(16)
G\c = 27s =—z—A"|C, 2/i
(17)
Using the relation
we obtain K,lllcd
/(I -»Vy \11/2 '1
—
ly
""')„ Klc = 0357Kic>0-350Kic.
(18)
This calculation suggests that screw dislocations cannot be nucleated spontaneously below the critical condition for cleavage, which is consistent with the expectation that oc-Fe single crystals be intrinsically brittle. However, the small difference between the numerical factors is most likely to be below the accuracy of the calculation, which viciates the argument to a considerable extent. Indeed, consideration of tension softening, the effect of anisotropy and uncertainties in the material parameters can all change eqn. (18) to some degree. The calculation does nevertheless provide a first indication that screw dislocation nucleation from a ledge may indeed be much easier than nucleation on inclined and oblique planes. Next we consider a semi-infinite crack under simple mode III loading. Tension softening has been shown to be of little consequence up to values of A'| of the order of 0-9Klc (Xu et al. 1995a) and can therefore be safely neglected. Figures 13 and 14 show two saddle-point configurations of the embryonic screw dislocations for norFig. 12
Distribution of stress intensity factors on a cleavage ledge.
Dislocation nucleation from crack lips
357
Fig. 13 Gin/GlltaP0.75
The saddle-point configuration for nucleation of a screw dislocation from a cleavage ledge, Giii/Giiicd = 0-75. Fig. 14 Gm/G1Ilcd=0.50
The saddle-point configuration for nucleation of a screw dislocation from a cleavage ledge. Gin/G|ii«d = 0-50-
malized load levels Gin/Gmcd = 0-75 and 0-50. Interestingly, these saddle-point configurations are flatter than those of edge dislocations, shown in figs. 3 and 4, as befits the lower line energies of screw dislocations. As the screw dislocation bows out, it tends to form double kinks with short edge components. By way of contrast, the screw double kinks of the edge dislocation embryo tend to be longer and the edge segment shorter (figs. 3 and 4). Computationally, this requires a larger periodic domain in the case of the screw embryo, which inevitably increases the size of the problem. The dependence of the activation energy on the crack driving force is
358
G. Xu et al.
shown in fig. 15. The low values of the activation energy relative to those computed for nucleation on inclined and oblique planes is particularly noteworthy. Finally, we endeavour to ascertain the magnitude of the errors incurred as a result of the various simplifications adopted in the calculations. In order to estimate the effect of tension softening, we consider the simple two-dimensional problem of a semi-infinite crack subjected to mixed mode I and III. We take the ratio Kt]]/Kt to be 0-35/0-81 = 0-43, which is the case of interest in the ledge problem. We also wish to estimate the effect of identifying the interlayer inelastic displacements with the total displacements, that is of setting A — 8, a simplification which has been adopted for computational convenience. Physically, this corresponds to taking the interplanar distance h across the slip plane to be zero. In the ledge problem, we have additionally set the parameter p = 0 for want of a better estimate. Figure 16 shows the effect on the activation energy of variations in these parameters. As is evident from the figure, the h = 0 approximation accounts for modest errors of the order of 15% at most, over much of the range of G|/C7|C. The effect of tension softening in the range O^A'i sg043KjH is indeed seen to be negligible. The lowest curve for the activation energy has been calculated for/; = 0-217 as a reasonable estimate. It shows that nonzero values of/; can reduce the activation energy significantly, which indicates the need for more reliable estimates of this parameter. The effect of the periodicity of the computational model on the activation energy was found by Xu et al. (1995a) to decay rapidly with increasing size of the period. 4.5. Estimates of the brittle-to-ductile transition temperature As noted in the foregoing, the B-D transition in bcc transition metals, and particularly in oc-Fe, is most likely probably controlled solely by dislocation nucleation. Therefore, the preceding results can be used to estimate the B-D transition temperatures attendant to the three nucleation modes considered, namely nucleation on inclined planes, on oblique planes, or on cleavage ledges. No precise experimental Fig. 15 i ■ ' ■ ' i
4
0.5
0.6
0.7
0.8
0.9
1.0
"llir'lllcd
The activation energy for dislocation emission in simple mode III loading, at the cleavage ledge.
Dislocation nucleation from crack tips
359
Fig. 16
DQl .... I . ... I . .
t>.4
0.5
0.6
0.7
0.8
0.9
G/GIC
Parametric sensitivity of the activation energy for dislocation emission on cleavage ledges.
measurements of the transition temperature of single-crystal oc-Fe are available. The transition temperature for polycrystalline low-C steel is about 250 K, as determined from Charpy impact experiments (McClintock and Argon 1966). In the absence of more direct measurements, we shall suppose the transition temperature for pure oc-Fe to be in the range 250-300 K. A B-D transition scenario achievable experimentally was proposed by Argon (1987) and consists of the arrest of a cleavage crack propagating against a temperature gradient. The evaluation of the B-D transition temperature from the activation energy can be effected as suggested by Xu et al. (1995a), who give the relation Ta.
[
BD
(19)
Here T0 = ßb3/k(l - v) « 1-2 x 105 K; the melting temperature Tm = 1809 K for ocFe; a = (1 -i/)AUacl/pb3 is the normalized activation energy; c is the speed of sound; t>« lcms-1 is a typical crack propagation velocity, giving ln(c/t>) « 10; rj « 0-5 is a coefficient describing the temperature dependence of the shear modulus which, to a first approximation, is presumed of the form (i = Po
(-'0
(20)
The dependence of the activation energy for nucleation of dislocation embryos on the energy release rate Gx is shown in fig. 17 for each of the three modes of nucleation considered in the foregoing. The activation energy at the critical driving force for cleavage, that is, at Gi/G|c = 1, determines the transition temperature through eqn. (19). This relation is plotted in fig. 18, together with the activation energies for nucleation on inclined planes, oblique planes and ledges. Also shown in the figure is the value of the transition temperature for polycrystalline Fe. It is evident from this comparison that only nucleation on cleavage ledges results in transition temperatures which approach the expected value for oc-Fe. The remaining two mechan-
360
G. Xu et al. Fig. 17
\ oblique plane ja ■^
4
inclined plane cleavage ledge
%0
The activation energies for dislocation nucleation at a crack tip in a-Fe, for three different modes of nucleation.
Fig. 18 i
"i
Tm 10J
TBD (poly-Fe)
10= / io!irr
i
irj1
§
a. u 3
.5
8> —
/
4)
c « a, *o u
.2" o
"o .E
'
lä1^
,
Tö°io1
mul .
io*
The estimated B-D transition temperatures in a-Fe.
isms grossly overestimate the transition temperature. These results strongly suggest that dislocation nucleation from a crack tip in a-Fe is an inhomogeneous process. The dislocation loops which eventually shield the crack are emitted from ledges distributed along the crack front. § 5. DISCUSSION We have viewed the transition in fracture behaviour from ductile to brittle or, more fundamentally, from brittle to ductile as a manifestation of crack-tip-initiated plasticity counteracting the tendency for brittle behaviour by cleavage cracking. In this regard, we have distinguished two different behaviours: that in which the transi-
Dislocation nucleat ion from crack tips
361
tion is controlled by nucleation of dislocation embryos from the crack tip, which is characteristic of bcc transition metals, and that in which the transition is controlled by the mobility of dislocations away from the crack tip, which is typical of semiconductors and compounds. The distinguishing characteristic between .these two behaviours is the mobility of kinks on dislocations. Experiments indicate that in bcc transition metals (at anything but the lowest temperatures) there is little resistance to the motion of kinks along dislocation lines. By contrast, in semiconductors such as Si and other compounds for which good information on dislocation mobility exists (Yonenaga et al. 1987, 1989, Yonenaga and Sumino 1989) it is known that the stress dependence of the dislocation velocity is nearly linear, implying that a process of kink drift controls dislocation motion. Computer simulations (Bulatov et al. 1995) confirm that kink motion along dislocations is indeed hindered by very substantial energy barriers in Si. These observations suggest that the B-D transition is nucleation controlled in bcc transition metals and mobility controlled in semiconductors and semiconducting compounds. An additional mechanism which can influence fracture behaviour is crack-tip shielding by general 'background' plasticity. A particularly elegant and compelling analysis of this mechanism was advanced by Freund and Hutchinson (1985). Based on the known rate dependence characteristics of steels, which exhibit a marked stress upturn at high strain rates, Freund and Hutchinson (1985) demonstrated that brittle fracture can take place at high crack propagation velocities. This results in progressively diminishing inelastic response but that the transition is smooth and spread out, and far from being abrupt. The importance of background plasticity effects has been demonstrated experimentally by Hirsch et al. (1989a), who have shown that the sharp B-D transition in dislocation-free Si becomes diffuse, and moves to somewhat lower temperatures, when the crystals are initially dislocated by a pre-deformation step. The effect of background plasticity can therefore be regarded as one of modulating the B-D transition, with the ultimate controlling mechanism residing in crack-tip-initiated processes. It is observed that intrinsically brittle materials such as crack-free Fe (Allen 1959) and W (Argon and Maloof 1966a) single crystals can often be plastically deformed at cryogenic temperatures with low plastic resistance when they are in pure form but fracture in a brittle manner when they are less pure and exhibit a high plastic resistance. This observation has been used as an argument in support of the background plasticity model of the fracture transition. This is partly correct. An intrinsically brittle solid only demonstrates its brittle characteristics when a crack is present. In well prepared Fe and W crystals, cracks do not exist initially and need to be produced by plastic deformation, most prominently by the intersection of deformation twins (Argon and Maloof 1966b). This requires a relatively high stress to nucleate the twins. In pure metallic crystals, the plastic yield stress is usually below the critical stress required for twinning, and brittle microcracks are not produced until the flow stress is raised by strain hardening to a sufficient level for twinning. In the present paper we have reported on the key process of dislocation nucleation from crack tips in bcc transition metals, and particularly in a-Fe, where nucleation is expected be the controlling process. We have noted that, as is now well established (Schock and Püschl 1991, Rice et al. 1992, Rice and Beltz 1994, Xu et al. 1995a), at the saddle point the critical activation configuration of the nucleated dislocation consists only of partially completed core matter. Consequently, we have termed these configurations dislocation embryos.
362
G. Xu et al.
We note that a cursory examination of the shapes of the dislocation embryos shown in figs. 3 and 4, for the case of the inclined plane, and to a lesser extent, those in figs. 13 and 14, for the substantially screw-type embryos on cleavage ledges might give the counter-intuitive appearance of closed loops that are in the process of formation. This is an illusion since the level contours shown in these figures pertain to the total displacements at the crack tip where only the bulged portions of these, in the region of xxjb > 0, could be associated with the inelastic displacements of the embryo. A reference line passed through xt/b = 0 (the initial geometrical crack front) shows clearly that the embryo approximates to somewhat less than a halfellipse but results in the partial penetration of the crack tip displacements into the embryo, giving an overall mushroom-type appearance. The present study, as was our original work, as well as that of Rice et al. (1992), is based on the Peierls concept of describing the fundamental inelastic response by an interplanar tension-shear potential. The analysis of the shapes of the embryos and the dependence of the activation energies for their formation on the applied energy release rates were performed self-consistently, utilizing quantities such as /i, 7US and 7S that had been determined by the best available atomistic approaches (Sun, Beltz and Rice 1993) and were presented also by us earlier (Xu et al. 1995a). Thus, while published information of these quantities based on experimental measurements may have a considerable latitude in certainty, our results should have greater accuracy in the relative placement of the modes of response that we have simulated. Clearly, however, on an absolute basis, such as in the determination of the transition temperatures of fig. 18, the results should be viewed with more caution. Nevertheless, the energetics of the embryos that we have analysed indicate that the very large uncertainties in the transition temperatures of previous considerations (discussed by Xu et al. (1995a)) have now been eliminated. In our present study we have examined three plausible modes of nucleation: on inclined planes containing the crack front, on oblique planes intersecting the crack front, in the interior and near a free surface and, finally, on cleavage ledges along the crack front. Our analysis has confirmed our earlier finding (Xu et al. 1995a) that nucleation on inclined planes in oc-Fe entails energy barriers that are too high to be overcome, at impending crack advance, at temperatures below the melting point. Contrary to expectations, our analysis has also established that dislocation nucleation on oblique planes in a-Fe requires even higher energies in the interior of the solid, which translates into transition temperatures well above the melting point. Since in this mode of nucleation no significant free surface is produced, the finding is surprising but can be explained by noting that, while the peak shear stress near the crack tip is higher on the oblique planes than on inclined planes, the area-averaged shear stress is significantly lower in the former case, owing to the rapid decay of stresses in all directions away from the tip. A rather different conclusion was reached, however, for oblique planes near a free surface where no plane-strain stress exists and shear stresses on oblique planes are much higher. Here, were it not for a need of some surface ledge production, emission of dislocations should be nearly spontaneous. Such preponderance of dislocation emission where the crack reaches free surfaces has been observed by George and Michot (1993). Both the inclined plane modes and the oblique plane modes are instances of homogeneous nucleation, inasmuch as every segment of the crack front constitutes an equally likely nucleation site. However, numerous experiments (Chiao and Clarke 1989, Samuels and Roberts 1989, George and Michot 1993) have demonstrated that
Dislocation nucleation from crack tips
363
nucleation is a rare event and occurs only at particular sites along the crack front. These sites are nearly always associated with cleavage ledges. A preliminary analysis of heterogeneous nucleation at ledges had been carried out by Zhou and Thomson (1991), who found the mechanism to be quite favourable. We have analysed dislocation nucleation on cleavage ledges in oc-Fe, where we have taken the crack front to coincide with the (110) direction and the ledges to be on {112} planes. This mode is favoured in two important ways. Firstly, the embryo is of a predominantly screw type and, hence, has a low line energy; it involves no surface production. Indeed, our results show that the energetics of this mode in
